Uniform boundedness principle

In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.

Theorem

Uniform Boundedness Principle — Let X be a Banach space and Y a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. If

\sup \nolimits _{T\in F}\|T(x)\|_{Y}<\infty ,

for all $x \in X$ , then

\sup \nolimits _{T\in F,\|x\|=1}\|T(x)\|_{Y}=\sup \nolimits _{T\in F}\|T\|_{B(X,Y)}<\infty .

The completeness of X enables the following short proof, using the Baire category theorem.

Proof

Let $X$ be a Banach space. Suppose that for every $x \in X$ ,

\sup \nolimits _{T\in F}\|T(x)\|_{Y}<\infty .

For every integer $n\in \mathbb {N}$ , let

{\displaystyle X_{n}=\left\{x\in X\

Each set $X_{n}$ is a closed set and by the assumption,

\bigcup \nolimits _{n\in \mathbf {N} }X_{n}=X\neq \varnothing .

By the Baire category theorem for the non-empty complete metric space X, there exists some $m$ such that $X_{m}$ has non-empty interior; that is, there exist $x_{0}\in X_{m}$ and $ε > 0$ such that

{\overline {B_{\varepsilon }(x_{0})}}:=\{x\in X\,:\,\|x-x_{0}\|\leq \varepsilon \}\subseteq X_{m}.

Let $u \in X$ with $ǁ u ǁ \leq 1$ and $T \in F$ . One has that:

{\begin{aligned}\|T(u)\|_{Y}&=\varepsilon ^{-1}\left\|T\left(x_{0}+\varepsilon u\right)-T(x_{0})\right\|_{Y}&[{\text{by linearity of }}T]\\&\leq \varepsilon ^{-1}\left(\left\|T(x_{0}+\varepsilon u)\right\|_{Y}+\left\|T(x_{0})\right\|_{Y}\right)\\&\leq \varepsilon ^{-1}(m+m).&[{\text{since}}\ x_{0}+\varepsilon u,\ x_{0}\in X_{m}]\\\end{aligned}}

Taking the supremum over u in the unit ball of X and over $T \in F$ it follows that

\sup \nolimits _{T\in F}\|T\|_{B(X,Y)}\leq 2\varepsilon ^{-1}m<\infty .

There are also simple proofs not using the Baire theorem (Sokal 2011).

Corollaries

Corollary — If a sequence of bounded operators $(T n)$ converges pointwise, that is, the limit of ${T n (x) }$ exists for all $x \in X$ , then these pointwise limits define a bounded operator T.

The above corollary does not claim that $T n$ converges to T in operator norm, i.e. uniformly on bounded sets. However, since ${T n}$ is bounded in operator norm, and the limit operator T is continuous, a standard "3-ε" estimate shows that $T n$ converges to T uniformly on compact sets.

Corollary — Any weakly bounded subset S in a normed space Y is bounded.

Indeed, the elements of S define a pointwise bounded family of continuous linear forms on the Banach space $X = Y*$ , continuous dual of Y. By the uniform boundedness principle, the norms of elements of S, as functionals on X, that is, norms in the second dual $Y**$ , are bounded. But for every $s \in S$ , the norm in the second dual coincides with the norm in Y, by a consequence of the Hahn–Banach theorem.

Let $L (X, Y)$ denote the continuous operators from X to Y, with the operator norm. If the collection F is unbounded in $L (X, Y)$ , then by the uniform boundedness principle, we have:

{\displaystyle R=\left\{x\in X\

In fact, R is dense in X. The complement of R in X is the countable union of closed sets $\cup X n$ . By the argument used in proving the theorem, each $X n$ is nowhere dense, i.e. the subset $\cup X n$ is of first category. Therefore R is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called residual sets) are dense. Such reasoning leads to the principle of condensation of singularities, which can be formulated as follows:

Theorem — Let X be a Banach space, ${Y n}$ a sequence of normed vector spaces, and $F n$ an unbounded family in $L (X, Y n)$ . Then the set

{\displaystyle R=\left\{x\in X\

is a residual set, and thus dense in X.

Proof

The complement of R is the countable union

{\displaystyle \bigcup \nolimits _{n,m}\left\{x\in X\

of sets of first category. Therefore, its residual set R is dense.

Example: pointwise convergence of Fourier series

Let $\mathbb {T}$ be the circle, and let $C(\mathbb {T} )$ be the Banach space of continuous functions on $\mathbb {T} ,$ with the uniform norm. Using the uniform boundedness principle, one can show that there exists an element in $C(\mathbb {T} )$ for which the Fourier series does not converge pointwise.

For $f\in C(\mathbb {T} ),$ its Fourier series is defined by

\sum _{k\in \mathbf {Z} }{\hat {f}}(k)e^{ikx}=\sum _{k\in \mathbf {Z} }{\frac {1}{2\pi }}\left(\int _{0}^{2\pi }f(t)e^{-ikt}dt\right)e^{ikx},

and the N-th symmetric partial sum is

S_{N}(f)(x)=\sum _{k=-N}^{N}{\hat {f}}(k)e^{ikx}={\frac {1}{2\pi }}\int _{0}^{2\pi }f(t)D_{N}(x-t)\,dt,

where D_N is the N-th Dirichlet kernel. Fix $x\in \mathbb {T}$ and consider the convergence of {S_N(f)(x)}. The functional $\varphi _{N,x}:C(\mathbb {T} )\to \mathbb {C}$ defined by

\varphi _{N,x}(f)=S_{N}(f)(x),\qquad f\in C(\mathbb {T} ),

is bounded. The norm of $φ N, x$ , in the dual of $C(\mathbb {T} )$ , is the norm of the signed measure $(2π) -1 D N (x - t) d t$ , namely

\left\|\varphi _{N,x}\right\|={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(x-t)\right|\,dt={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(s)\right|\,ds=\left\|D_{N}\right\|_{L^{1}(\mathbb {T} )}.

One can verify that

{\frac {1}{2\pi }}\int _{0}^{2\pi }|D_{N}(t)|\,dt\geq {\frac {1}{2\pi }}\int _{0}^{2\pi }{\frac {\left|\sin \left((N+{\tfrac {1}{2}})t\right)\right|}{t/2}}\,dt\to \infty .

So the collection ${ φ N, x}$ is unbounded in $C(\mathbb {T} )^{\ast },$ the dual of $C(\mathbb {T} ).$ Therefore, by the uniform boundedness principle, for any $x\in \mathbb {T} ,$ the set of continuous functions whose Fourier series diverges at x is dense in $C(\mathbb {T} ).$

More can be concluded by applying the principle of condensation of singularities. Let ${x m}$ be a dense sequence in $\mathbb {T} .$ Define $φ N, x m$ in the similar way as above. The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each $x m$ is dense in $C(\mathbb {T} )$ (however, the Fourier series of a continuous function f converges to $f (x)$ for almost every $x\in \mathbb {T}$ , by Carleson's theorem).

Generalisations

The least restrictive setting for the uniform boundedness principle is a barrelled space where the following generalized version of the theorem holds (Bourbaki 1987, Theorem III.2.1):

Theorem — Given a barrelled space X and a locally convex space Y, then any family of pointwise bounded continuous linear mappings from X to Y is equicontinuous (even uniformly equicontinuous).

Alternatively, the statement also holds whenever X is a Baire space and Y is a locally convex space.[1]

Dieudonné (1970) proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces. Specifically,

Theorem — Let X be a Fréchet space, Y a normed space, and H a set of continuous linear mappings of X into Y. If for every $x \in X$ ,

\sup \nolimits _{u\in H}\|u(x)\|<\infty

then the family H is equicontinuous.

Citations

Shtern 2001.

Bibliography

Banach, Stefan; Steinhaus, Hugo (1927), "Sur le principe de la condensation de singularités" (PDF), Fundamenta Mathematicae, 9: 50–61, doi:10.4064/fm-9-1-50-61. (in French)
Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
Dieudonné, Jean (1970), Treatise on analysis, Volume 2, Academic Press.
Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
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Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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Sokal, Alan (2011), "A really simple elementary proof of the uniform boundedness theorem", Amer. Math. Monthly, 118 (5): 450–452, arXiv:1005.1585, doi:10.4169/amer.math.monthly.118.05.450, S2CID 41853641.
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[FOOTNOTEShtern2001-1] Shtern 2001.

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results	Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)
Maps	Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property